Applied Analysis REU
Research Experiences for Undergraduates
June 2 - July 27, 2024
Research Experiences for Undergraduates
June 2 - July 27, 2024
All the carefully designed projects are in the area of applied mathematics, emphasizing the applications of the mathematical tools that an undergraduate student can master.
Projects are computational, theoretical, or a combination of these. The required techniques are mostly covered in calculus and differential equations undergraduate courses.
The participants present their work at the WVU Undergraduate Research Symposium, an annual summer event for undergraduate students of the region. In addition, a graduate student of the School of Mathematical and Data Sciences assists with this REU program and provides information needed for graduate school life.
Faculty Mentor: Charis Tsikkou
Conservation laws, which have important applications in gas dynamics, fluid mechanics, elasticity, aircraft aerodynamics, geophysical fluid dynamics, meteorology, general relativity, bio-sciences etc, state that the total amount of a certain quantity, such as mass, momentum, or energy, will not change in time during physical processes and can be written as a system of PDEs in divergence form. More generally, we consider systems of balance laws, which stipulate that the time rate of change in the amount of a quantity inside a domain is balanced by the rate of flux of this quantity through the domain’s boundary together with the rate of its production inside the domain.
The global-in-time existence of solutions to an initial value problem (Cauchy problem) is of particular interest. A fundamental feature of hyperbolic systems of balance laws is the development of discontinuities or shocks (infinite gradient) in finite time, even starting from smooth initial values. Because of this, solutions in the classical sense could fail to exist.
When studying multi-dimensional problems, it is common to consider simpler solutions with planar, cylindrical, and spherical symmetry, which appear in systems modeling the early stages of a gas flow between fragments of a bomb explosion with a spherical or cylindrical casing, nuclear explosions, etc. Seeking symmetric solutions has the advantage of changing a multi-d problem to a 1-dimensional one. Another common practice is seeking self-similar solutions, a common viewpoint in fluid mechanics. At this stage, we pursue a theory of the existence of solutions for a simpler class of initial data to models used in applications to develop methods that can provide insight into the stability and behavior of solutions. Such results can be used as building blocks in generalized schemes when solving Cauchy problems globally with large data.
The basic building block towards the proof of existence theorems of the Cauchy problem of conservation laws in one space dimension with small data is the solution to the Riemann problem, an initial value problem that consists of data containing two constant states, separated by a discontinuity at the origin which gives self-similar solutions. The solution is synthesized by constant states, shocks joining constant states that satisfy the Rankine-Hugoniot relation and/or centered rarefaction waves bordered by constant states or contact discontinuities. The main focus of the projects on this topic is the study of models whose solutions to the Riemann problems involve the so-called singular shocks, which are a more compressive generalization of the ordinary shock wave where at least one state variable develops an extreme concentration in the form of a weighted Dirac delta function. Constructing Riemann solutions is essential from both analytical and numerical points of view.
Faculty Mentor: Adrian Tudorascu
In the early 1970’s, Zeldovich proposed a basic particles model describing the formation of large structures in the universe by accretion of matter. If the initial distribution of mass is discrete and supported at finitely many points, then the particle evolution is constrained by the following laws: between collisions, each particle moves along the line at constant velocity. Particles involved in a collision stick together (perfectly inelastic collisions) and form another particle whose mass is the total mass participating in the collision. The velocity of the newly formed particle is given by the conservation of momentum. Zeldovich also noticed that the particle distribution and its velocity satisfy the so-called Pressureless Euler system. It was not until two decades later that the first mathematical results on existence of solutions appeared. These results were also only proved under very restrictive conditions on the initial distribution and velocity. Since the distributions involved are generically singular measures (rather than functions), the progress on this problem was slow and incremental until powerful tools from the Optimal Transport theory weighed in and sped up progress. By now we have a fairly good understanding of the well-posedness theory, even though some restrictions on the initial velocity have not so far been removed.
Very recently, Dr. Tudorascu proposed and performed a first analysis of the case where the particles are confined to a closed subset of the real line. Here one needs to deal with boundary conditions and devise a notion of solution for general distributions, which guarantees that when the initial distribution is a discrete measure then the particles reaching the boundary are trapped there. In fact, this trapping boundary is a toy model for a Black Hole.
Interestingly, when the evolution is confined to compact subsets of the line, the particle distribution has an asymptotic limit, in other words the system tends to settle into an equilibrium of stationary particles (some on the boundary). The tough part seems to be quantifying the rate of decay to the equilibrium; it will be interesting to use numerics in order to propose an educated guess as to what that rate should be. Can we then prove that that is the correct rate?
Research on the sticky particles model will enhance our comprehension of the formation of large-scale structures in the universe because more complex and realistic models (with attraction or repulsion between the particles) are amenable to similar techniques.
Note, additional projects may be added to this list. If you have a special request, please contact the directors for help.